Novel Algorithms for Nonlinear Optimization

本项目的两个研究目标是发展、分析和实施新的数值方法来解决非线性优化问题。第一个重点是非线性规划（NLP）方法的设计，与现有算法相比，它可以重用导数矩阵的因式分解来解决密切相关的问题实例。这种热启动方法有望显著提高混合整数非线性优化的分支定界算法的速度。第二个重点是基于广义Benders分解的高效并行算法的发展，用于解决可分解nlp，因为它们出现在不确定设计或两阶段随机优化问题中。重点在于非凸问题的局部解的快速计算，而现有的方法仅限于凸实例或限于更耗时的全局最优搜索。数值优化已经成为工业，经济和科学许多领域不可或缺的工具，回答诸如“设计和运行该工厂的最佳方法是什么”或“如何运行电网以便能够承受网络组件的故障”等问题。虽然强大的计算方法可用于系统的优化，这些系统可以用线性或限制于非离散决策的模型来描述，但在实践中经常出现的非线性和离散问题的解决方案，在当前技术下往往过于耗时。因此，提出的研究项目的第一部分旨在显著加快非线性离散优化算法的关键组成部分。该项目的第二个研究目标是有效利用日益普及的并行计算能力，以优化考虑许多潜在场景的问题，作为解决未来环境不确定性的一种方式。

The two research objectives of this project concern the development, analysis, and implementation of novel numerical methods for the solution of nonlinear optimization problems. The first thrust addresses the design of nonlinear programming (NLP) methods that can, in contrast to existing algorithms, reuse the factorization of derivative matrices for the solution of closely related problem instances. Such hot-started methods are expected to lead to significant speedup of branch-and-bound algorithms for mixed-integer nonlinear optimization. The second focus is the development of efficient parallel algorithms based on Generalized Benders Decomposition for the solution of decomposable NLPs, as they arise in design under uncertainty or two-stage stochastic optimization problems. The emphasis lies in the fast computation of local solutions of nonconvex problems, whereas existing approaches are restricted to convex instances or limited to the much more time-consuming search for global optima.Numerical optimization has become an indispensable tool in many areas of industry, economy and science, answering questions such as "what is the best way to design and operate this plant" or "how should the electrical power grid be operated in order to be able to sustain failure of network components." While powerful computational methods are available for the optimization of systems that can be described by models that are either linear or restricted to non-discrete decisions, the solution of problems that are both nonlinear and discrete, as they frequently appear in practice, is often too time-consuming with current technology. Therefore, the first part of the proposed research project aims at significantly accelerating a crucial key component in algorithms for nonlinear discrete optimization. The second research objective of this project deals with the efficient exploitation of increasingly pervasive parallel computing power for the optimization of problems that consider many potential scenarios as a way of addressing the uncertainty of future circumstances.